Today s class. Roots of equation Finish up incremental search Open methods. Numerical Methods, Fall 2011 Lecture 5. Prof. Jinbo Bi CSE, UConn

Transcription

1 Today s class Roots of equation Finish up incremental search Open methods 1

2 False Position Method Although the interval [a,b] where the root becomes iteratively closer with the false position method, unlike the bisection method, the size of the interval does not necessarily converge to zero. Sometimes it can cause the false position to converge slower than bisection Lecture 4 2

3 False Position Method Lecture 4 3

4 False Position Method Modified False Position Method Detect when you get stuck and use a bisection method Can get you to convergence faster Lecture 4 4

5 Incremental Searches Dependent on knowing the bracket in which the root falls Can use bracketed incremental search to speed up exhaustive search How big a bracket or increment can determine how long the search will take Too small increment and it will take too long Too big increment may miss roots, in partular, the multiple roots Lecture 4 5

6 Incremental Searches Lecture 4 6

7 Open Methods Bracket methods depend on knowing the interval in which the root resides What if you don t know the upper and lower bound on the root? Open methods Use a single estimate of the root Use two starting points but not bracketing the root May not converge on root 7

8 Open Methods 8

9 Open Methods Fixed-Point Iteration One-point iteration Successive substitution Start with equation f(x) = 0 and rearrange so x is on left hand side. If algebraic manipulation doesn t work, just add x to both sides 9

10 Fixed-point iteration The function transformation allows us to use g(x) to calculate a new guess of x 10

11 Example Find root of f(x)=e -x -x Transform f(x)=0 to x=g(x)=e -x Start with an estimate of x 0 =0 x 1 =g(x 0 )=e -0 =1 11

12 Example true value of the root:

13 Example 13

14 Fixed-point iteration Convergence properties If converge, much faster than bracketing methods May not converge Depends on the curve characteristics 14

15 Fixed-point iteration 15

16 Fixed-point iteration 16

17 Fixed-point iteration 17

18 Fixed-point iteration 18

19 Convergence Analysis Assume x r is the true root Combine with the iterative relationship 19

20 Fixed-point iteration Use derivative mean-value theorem If the derivative is less than 1, the error will get smaller with each iteration (monotonic or oscillating). If the derivative is greater than 1, the error will get larger with each iteration. 20

21 Newton-Raphson Method Similar idea to False Position Method Use tangent to guide you to the root 21

22 Example Find root of f(x)=e -x -x Start with an estimate of x 0 =0 22

23 Example true value of the root:

24 Newton-Raphson Method Convergence analysis First-order Taylor series expansion At root 24

25 Newton-Raphson Method Newton-Raphson method is quadratically convergent E t, i+ 1 = f "( xr ) 2 f '( x ) r E 2 t, i 25

26 Newton-Raphson Method Problems and Pitfalls Slow convergence when initial guess is not close enough May not converge at all Problems with multiple roots 26

27 Newton-Raphson Method 27

28 Newton-Raphson Method 28

29 Newton-Raphson Method 29

30 Newton-Raphson Method 30

31 Newton-Raphson Method Algorithm should guard against slow convergence or divergence If slow convergence or divergence detected, use another method 31

32 Secant method Newton-Raphson method requires calculation of the derivative Instead, approximate the derivative using backward finite divided difference 32

33 Secant method From Newton-Raphson method Replace with backward finite difference approximation 33

34 Example Find root of f(x)=e -x -x Start with an estimate of x -1 =0 and x 0 =1 34

35 Example true value of the root:

36 Secant Method vs. False- Position Method False-Position method always brackets the root False-Position will always converge Secant method may not converge Secant method usually converges much faster 36

37 Secant Method vs. False- Position Method 37

38 Modified Secant Method Instead of using backward finite difference to estimate the derivative, use a small delta Substitute back into Newton-Raphson formula 38

39 Example Find root of f(x)=e -x -x Start with an estimate of x 0 =1 and δ=

40 Example true value of the root:

41 Next class Polynomial roots Read Chapter 7 41